TY - JOUR

T1 - Approximating conditional distribution functions using dimension reduction

AU - Hall, Peter

AU - Yao, Qiwei

PY - 2005/6

Y1 - 2005/6

N2 - Motivated by applications to prediction and forecasting, we suggest methods for approximating the conditional distribution function of a random variable Y given a dependent random d-vector X. The idea is to estimate not the distribution of Y|X, but that of Y|θ T X, where the unit vector θ is selected so that the approximation is optimal under a least-squares criterion. We show that θ may be estimated root-n consistently. Furthermore, estimation of the conditional distribution function of Y, given θ T X, has the same first-order asymptotic properties that it would enjoy if θ were known. The proposed method is illustrated using both simulated and real-data examples, showing its effectiveness for both independent datasets and data from time series. Numerical work corroborates the theoretical result that θ can be estimated particularly accurately.

AB - Motivated by applications to prediction and forecasting, we suggest methods for approximating the conditional distribution function of a random variable Y given a dependent random d-vector X. The idea is to estimate not the distribution of Y|X, but that of Y|θ T X, where the unit vector θ is selected so that the approximation is optimal under a least-squares criterion. We show that θ may be estimated root-n consistently. Furthermore, estimation of the conditional distribution function of Y, given θ T X, has the same first-order asymptotic properties that it would enjoy if θ were known. The proposed method is illustrated using both simulated and real-data examples, showing its effectiveness for both independent datasets and data from time series. Numerical work corroborates the theoretical result that θ can be estimated particularly accurately.

KW - Conditional distribution

KW - Cross-validation

KW - Dimension reduction

KW - Kernel methods

KW - Leave-one-out method

KW - Local linear regression

KW - Nonparametric regression

KW - Prediction

KW - Root-n consistency

KW - Time series analysis

UR - http://www.scopus.com/inward/record.url?scp=23744468033&partnerID=8YFLogxK

U2 - 10.1214/009053604000001282

DO - 10.1214/009053604000001282

M3 - Article

SN - 0090-5364

VL - 33

SP - 1404

EP - 1421

JO - Annals of Statistics

JF - Annals of Statistics

IS - 3

ER -